Integrand size = 37, antiderivative size = 284 \[ \int \sqrt {1-d x} \sqrt {1+d x} (e+f x)^2 \left (A+B x+C x^2\right ) \, dx=\frac {\left (2 C e^2+8 A d^2 e^2+4 B e f+2 A f^2+\frac {C f^2}{d^2}\right ) x \sqrt {1-d^2 x^2}}{16 d^2}-\frac {\left (2 B-\frac {C e}{f}\right ) (e+f x)^2 \left (1-d^2 x^2\right )^{3/2}}{10 d^2}-\frac {C (e+f x)^3 \left (1-d^2 x^2\right )^{3/2}}{6 d^2 f}+\frac {\left (8 \left (C \left (d^2 e^3-4 e f^2\right )-2 f \left (5 A d^2 e f+B \left (d^2 e^2+f^2\right )\right )\right )-3 f^2 \left (2 d^2 e \left (2 B-\frac {C e}{f}\right )+5 \left (C+2 A d^2\right ) f\right ) x\right ) \left (1-d^2 x^2\right )^{3/2}}{120 d^4 f}+\frac {\left (C \left (2 d^2 e^2+f^2\right )+2 d^2 \left (2 B e f+A \left (4 d^2 e^2+f^2\right )\right )\right ) \arcsin (d x)}{16 d^5} \] Output:
1/16*(2*C*e^2+8*A*d^2*e^2+4*B*e*f+2*A*f^2+C*f^2/d^2)*x*(-d^2*x^2+1)^(1/2)/ d^2-1/10*(2*B-C*e/f)*(f*x+e)^2*(-d^2*x^2+1)^(3/2)/d^2-1/6*C*(f*x+e)^3*(-d^ 2*x^2+1)^(3/2)/d^2/f+1/120*(8*C*(d^2*e^3-4*e*f^2)-16*f*(5*A*d^2*e*f+B*(d^2 *e^2+f^2))-3*f^2*(2*d^2*e*(2*B-C*e/f)+5*(2*A*d^2+C)*f)*x)*(-d^2*x^2+1)^(3/ 2)/d^4/f+1/16*(C*(2*d^2*e^2+f^2)+2*d^2*(2*B*e*f+A*(4*d^2*e^2+f^2)))*arcsin (d*x)/d^5
Time = 1.27 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.92 \[ \int \sqrt {1-d x} \sqrt {1+d x} (e+f x)^2 \left (A+B x+C x^2\right ) \, dx=\frac {d \sqrt {1-d^2 x^2} \left (10 A d^2 \left (12 d^2 e^2 x+16 e f \left (-1+d^2 x^2\right )+3 f^2 x \left (-1+2 d^2 x^2\right )\right )+4 B \left (-8 f^2-d^2 \left (20 e^2+15 e f x+4 f^2 x^2\right )+2 d^4 x^2 \left (10 e^2+15 e f x+6 f^2 x^2\right )\right )+C \left (30 d^2 e^2 x \left (-1+2 d^2 x^2\right )+32 e f \left (-2-d^2 x^2+3 d^4 x^4\right )+5 f^2 x \left (-3-2 d^2 x^2+8 d^4 x^4\right )\right )\right )+30 \left (C \left (2 d^2 e^2+f^2\right )+2 d^2 \left (2 B e f+A \left (4 d^2 e^2+f^2\right )\right )\right ) \arctan \left (\frac {d x}{-1+\sqrt {1-d^2 x^2}}\right )}{240 d^5} \] Input:
Integrate[Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)^2*(A + B*x + C*x^2),x]
Output:
(d*Sqrt[1 - d^2*x^2]*(10*A*d^2*(12*d^2*e^2*x + 16*e*f*(-1 + d^2*x^2) + 3*f ^2*x*(-1 + 2*d^2*x^2)) + 4*B*(-8*f^2 - d^2*(20*e^2 + 15*e*f*x + 4*f^2*x^2) + 2*d^4*x^2*(10*e^2 + 15*e*f*x + 6*f^2*x^2)) + C*(30*d^2*e^2*x*(-1 + 2*d^ 2*x^2) + 32*e*f*(-2 - d^2*x^2 + 3*d^4*x^4) + 5*f^2*x*(-3 - 2*d^2*x^2 + 8*d ^4*x^4))) + 30*(C*(2*d^2*e^2 + f^2) + 2*d^2*(2*B*e*f + A*(4*d^2*e^2 + f^2) ))*ArcTan[(d*x)/(-1 + Sqrt[1 - d^2*x^2])])/(240*d^5)
Time = 0.67 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.95, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.243, Rules used = {2112, 2185, 27, 687, 25, 27, 676, 211, 223}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sqrt {1-d x} \sqrt {d x+1} (e+f x)^2 \left (A+B x+C x^2\right ) \, dx\) |
| \(\Big \downarrow \) 2112 |
| \(\displaystyle \int \sqrt {1-d^2 x^2} (e+f x)^2 \left (A+B x+C x^2\right )dx\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle -\frac {\int -3 f (e+f x)^2 \left (\left (2 A d^2+C\right ) f-d^2 (C e-2 B f) x\right ) \sqrt {1-d^2 x^2}dx}{6 d^2 f^2}-\frac {C \left (1-d^2 x^2\right )^{3/2} (e+f x)^3}{6 d^2 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (e+f x)^2 \left (\left (2 A d^2+C\right ) f-d^2 (C e-2 B f) x\right ) \sqrt {1-d^2 x^2}dx}{2 d^2 f}-\frac {C \left (1-d^2 x^2\right )^{3/2} (e+f x)^3}{6 d^2 f}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\frac {1}{5} \left (1-d^2 x^2\right )^{3/2} (e+f x)^2 (C e-2 B f)-\frac {\int -d^2 (e+f x) \left (f \left (10 A e d^2+3 C e+4 B f\right )+\left (5 \left (2 A d^2+C\right ) f^2-2 d^2 e (C e-2 B f)\right ) x\right ) \sqrt {1-d^2 x^2}dx}{5 d^2}}{2 d^2 f}-\frac {C \left (1-d^2 x^2\right )^{3/2} (e+f x)^3}{6 d^2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int d^2 (e+f x) \left (f \left (10 A e d^2+3 C e+4 B f\right )+\left (5 \left (2 A d^2+C\right ) f^2-2 d^2 e (C e-2 B f)\right ) x\right ) \sqrt {1-d^2 x^2}dx}{5 d^2}+\frac {1}{5} \left (1-d^2 x^2\right )^{3/2} (e+f x)^2 (C e-2 B f)}{2 d^2 f}-\frac {C \left (1-d^2 x^2\right )^{3/2} (e+f x)^3}{6 d^2 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \int (e+f x) \left (f \left (10 A e d^2+3 C e+4 B f\right )+\left (5 \left (2 A d^2+C\right ) f^2-2 d^2 e (C e-2 B f)\right ) x\right ) \sqrt {1-d^2 x^2}dx+\frac {1}{5} \left (1-d^2 x^2\right )^{3/2} (e+f x)^2 (C e-2 B f)}{2 d^2 f}-\frac {C \left (1-d^2 x^2\right )^{3/2} (e+f x)^3}{6 d^2 f}\) |
| \(\Big \downarrow \) 676 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {5 f \left (2 d^2 \left (A \left (4 d^2 e^2+f^2\right )+2 B e f\right )+C \left (2 d^2 e^2+f^2\right )\right ) \int \sqrt {1-d^2 x^2}dx}{4 d^2}+\frac {1}{4} f x \left (1-d^2 x^2\right )^{3/2} \left (-10 A f^2-4 B e f-\frac {5 C f^2}{d^2}+2 C e^2\right )+\frac {2 \left (1-d^2 x^2\right )^{3/2} \left (-10 A d^2 e f^2-2 B d^2 e^2 f-2 B f^3+C d^2 e^3-4 C e f^2\right )}{3 d^2}\right )+\frac {1}{5} \left (1-d^2 x^2\right )^{3/2} (e+f x)^2 (C e-2 B f)}{2 d^2 f}-\frac {C \left (1-d^2 x^2\right )^{3/2} (e+f x)^3}{6 d^2 f}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {5 f \left (2 d^2 \left (A \left (4 d^2 e^2+f^2\right )+2 B e f\right )+C \left (2 d^2 e^2+f^2\right )\right ) \left (\frac {1}{2} \int \frac {1}{\sqrt {1-d^2 x^2}}dx+\frac {1}{2} x \sqrt {1-d^2 x^2}\right )}{4 d^2}+\frac {1}{4} f x \left (1-d^2 x^2\right )^{3/2} \left (-10 A f^2-4 B e f-\frac {5 C f^2}{d^2}+2 C e^2\right )+\frac {2 \left (1-d^2 x^2\right )^{3/2} \left (-10 A d^2 e f^2-2 B d^2 e^2 f-2 B f^3+C d^2 e^3-4 C e f^2\right )}{3 d^2}\right )+\frac {1}{5} \left (1-d^2 x^2\right )^{3/2} (e+f x)^2 (C e-2 B f)}{2 d^2 f}-\frac {C \left (1-d^2 x^2\right )^{3/2} (e+f x)^3}{6 d^2 f}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {5 f \left (\frac {\arcsin (d x)}{2 d}+\frac {1}{2} x \sqrt {1-d^2 x^2}\right ) \left (2 d^2 \left (A \left (4 d^2 e^2+f^2\right )+2 B e f\right )+C \left (2 d^2 e^2+f^2\right )\right )}{4 d^2}+\frac {1}{4} f x \left (1-d^2 x^2\right )^{3/2} \left (-10 A f^2-4 B e f-\frac {5 C f^2}{d^2}+2 C e^2\right )+\frac {2 \left (1-d^2 x^2\right )^{3/2} \left (-10 A d^2 e f^2-2 B d^2 e^2 f-2 B f^3+C d^2 e^3-4 C e f^2\right )}{3 d^2}\right )+\frac {1}{5} \left (1-d^2 x^2\right )^{3/2} (e+f x)^2 (C e-2 B f)}{2 d^2 f}-\frac {C \left (1-d^2 x^2\right )^{3/2} (e+f x)^3}{6 d^2 f}\) |
Input:
Int[Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)^2*(A + B*x + C*x^2),x]
Output:
-1/6*(C*(e + f*x)^3*(1 - d^2*x^2)^(3/2))/(d^2*f) + (((C*e - 2*B*f)*(e + f* x)^2*(1 - d^2*x^2)^(3/2))/5 + ((2*(C*d^2*e^3 - 2*B*d^2*e^2*f - 4*C*e*f^2 - 10*A*d^2*e*f^2 - 2*B*f^3)*(1 - d^2*x^2)^(3/2))/(3*d^2) + (f*(2*C*e^2 - 4* B*e*f - 10*A*f^2 - (5*C*f^2)/d^2)*x*(1 - d^2*x^2)^(3/2))/4 + (5*f*(C*(2*d^ 2*e^2 + f^2) + 2*d^2*(2*B*e*f + A*(4*d^2*e^2 + f^2)))*((x*Sqrt[1 - d^2*x^2 ])/2 + ArcSin[d*x]/(2*d)))/(4*d^2))/5)/(2*d^2*f)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p + 3))/(c*(2*p + 3)) Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g , p}, x] && !LeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) ), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp [c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x ] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 0])
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f _.)*(x_))^(p_.), x_Symbol] :> Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; F reeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] & & EqQ[m, n] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) ^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Time = 0.48 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.26
| method | result | size |
| risch | \(-\frac {\left (40 C \,f^{2} x^{5} d^{4}+48 B \,d^{4} f^{2} x^{4}+96 C \,d^{4} e f \,x^{4}+60 A \,d^{4} f^{2} x^{3}+120 B \,d^{4} e f \,x^{3}+60 C \,d^{4} e^{2} x^{3}+160 A \,d^{4} e f \,x^{2}+80 B \,d^{4} e^{2} x^{2}+120 A \,d^{4} e^{2} x -10 C \,d^{2} f^{2} x^{3}-16 B \,d^{2} f^{2} x^{2}-32 C \,d^{2} e f \,x^{2}-30 A \,d^{2} f^{2} x -60 B \,d^{2} e f x -30 C \,d^{2} e^{2} x -160 A \,d^{2} e f -80 B \,d^{2} e^{2}-15 C \,f^{2} x -32 B \,f^{2}-64 C e f \right ) \sqrt {x d +1}\, \left (x d -1\right ) \sqrt {\left (-x d +1\right ) \left (x d +1\right )}}{240 d^{4} \sqrt {-\left (x d +1\right ) \left (x d -1\right )}\, \sqrt {-x d +1}}+\frac {\left (8 A \,d^{4} e^{2}+2 A \,d^{2} f^{2}+4 B \,d^{2} e f +2 C \,d^{2} e^{2}+C \,f^{2}\right ) \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right ) \sqrt {\left (-x d +1\right ) \left (x d +1\right )}}{16 d^{4} \sqrt {d^{2}}\, \sqrt {-x d +1}\, \sqrt {x d +1}}\) | \(358\) |
| default | \(\frac {\sqrt {-x d +1}\, \sqrt {x d +1}\, \left (120 A \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) d^{4} e^{2}+30 A \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) d^{2} f^{2}+30 C \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) d^{2} e^{2}-60 B \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d^{3} e f x +96 C \,\operatorname {csgn}\left (d \right ) d^{5} e f \,x^{4} \sqrt {-d^{2} x^{2}+1}+120 B \,\operatorname {csgn}\left (d \right ) d^{5} e f \,x^{3} \sqrt {-d^{2} x^{2}+1}+60 A \,\operatorname {csgn}\left (d \right ) d^{5} f^{2} x^{3} \sqrt {-d^{2} x^{2}+1}+60 C \,\operatorname {csgn}\left (d \right ) d^{5} e^{2} x^{3} \sqrt {-d^{2} x^{2}+1}-15 C \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d \,f^{2} x +60 B \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) d^{2} e f +120 A \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d^{5} e^{2} x -30 C \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d^{3} e^{2} x +80 B \,\operatorname {csgn}\left (d \right ) d^{5} e^{2} x^{2} \sqrt {-d^{2} x^{2}+1}-10 C \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d^{3} f^{2} x^{3}-16 B \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d^{3} f^{2} x^{2}-160 A \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d^{3} e f -64 C \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d e f +40 C \,\operatorname {csgn}\left (d \right ) d^{5} f^{2} x^{5} \sqrt {-d^{2} x^{2}+1}+48 B \,\operatorname {csgn}\left (d \right ) d^{5} f^{2} x^{4} \sqrt {-d^{2} x^{2}+1}-30 A \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d^{3} f^{2} x +160 A \,\operatorname {csgn}\left (d \right ) d^{5} e f \,x^{2} \sqrt {-d^{2} x^{2}+1}-32 C \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d^{3} e f \,x^{2}-80 B \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d^{3} e^{2}-32 B \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d \,f^{2}+15 C \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) f^{2}\right ) \operatorname {csgn}\left (d \right )}{240 \sqrt {-d^{2} x^{2}+1}\, d^{5}}\) | \(652\) |
Input:
int((-d*x+1)^(1/2)*(d*x+1)^(1/2)*(f*x+e)^2*(C*x^2+B*x+A),x,method=_RETURNV ERBOSE)
Output:
-1/240/d^4*(40*C*d^4*f^2*x^5+48*B*d^4*f^2*x^4+96*C*d^4*e*f*x^4+60*A*d^4*f^ 2*x^3+120*B*d^4*e*f*x^3+60*C*d^4*e^2*x^3+160*A*d^4*e*f*x^2+80*B*d^4*e^2*x^ 2+120*A*d^4*e^2*x-10*C*d^2*f^2*x^3-16*B*d^2*f^2*x^2-32*C*d^2*e*f*x^2-30*A* d^2*f^2*x-60*B*d^2*e*f*x-30*C*d^2*e^2*x-160*A*d^2*e*f-80*B*d^2*e^2-15*C*f^ 2*x-32*B*f^2-64*C*e*f)*(d*x+1)^(1/2)*(d*x-1)/(-(d*x+1)*(d*x-1))^(1/2)*((-d *x+1)*(d*x+1))^(1/2)/(-d*x+1)^(1/2)+1/16*(8*A*d^4*e^2+2*A*d^2*f^2+4*B*d^2* e*f+2*C*d^2*e^2+C*f^2)/d^4/(d^2)^(1/2)*arctan((d^2)^(1/2)*x/(-d^2*x^2+1)^( 1/2))*((-d*x+1)*(d*x+1))^(1/2)/(-d*x+1)^(1/2)/(d*x+1)^(1/2)
Time = 0.08 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.98 \[ \int \sqrt {1-d x} \sqrt {1+d x} (e+f x)^2 \left (A+B x+C x^2\right ) \, dx=\frac {{\left (40 \, C d^{5} f^{2} x^{5} - 80 \, B d^{3} e^{2} + 48 \, {\left (2 \, C d^{5} e f + B d^{5} f^{2}\right )} x^{4} - 32 \, B d f^{2} + 10 \, {\left (6 \, C d^{5} e^{2} + 12 \, B d^{5} e f + {\left (6 \, A d^{5} - C d^{3}\right )} f^{2}\right )} x^{3} - 32 \, {\left (5 \, A d^{3} + 2 \, C d\right )} e f + 16 \, {\left (5 \, B d^{5} e^{2} - B d^{3} f^{2} + 2 \, {\left (5 \, A d^{5} - C d^{3}\right )} e f\right )} x^{2} - 15 \, {\left (4 \, B d^{3} e f - 2 \, {\left (4 \, A d^{5} - C d^{3}\right )} e^{2} + {\left (2 \, A d^{3} + C d\right )} f^{2}\right )} x\right )} \sqrt {d x + 1} \sqrt {-d x + 1} - 30 \, {\left (4 \, B d^{2} e f + 2 \, {\left (4 \, A d^{4} + C d^{2}\right )} e^{2} + {\left (2 \, A d^{2} + C\right )} f^{2}\right )} \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{240 \, d^{5}} \] Input:
integrate((-d*x+1)^(1/2)*(d*x+1)^(1/2)*(f*x+e)^2*(C*x^2+B*x+A),x, algorith m="fricas")
Output:
1/240*((40*C*d^5*f^2*x^5 - 80*B*d^3*e^2 + 48*(2*C*d^5*e*f + B*d^5*f^2)*x^4 - 32*B*d*f^2 + 10*(6*C*d^5*e^2 + 12*B*d^5*e*f + (6*A*d^5 - C*d^3)*f^2)*x^ 3 - 32*(5*A*d^3 + 2*C*d)*e*f + 16*(5*B*d^5*e^2 - B*d^3*f^2 + 2*(5*A*d^5 - C*d^3)*e*f)*x^2 - 15*(4*B*d^3*e*f - 2*(4*A*d^5 - C*d^3)*e^2 + (2*A*d^3 + C *d)*f^2)*x)*sqrt(d*x + 1)*sqrt(-d*x + 1) - 30*(4*B*d^2*e*f + 2*(4*A*d^4 + C*d^2)*e^2 + (2*A*d^2 + C)*f^2)*arctan((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/ (d*x)))/d^5
\[ \int \sqrt {1-d x} \sqrt {1+d x} (e+f x)^2 \left (A+B x+C x^2\right ) \, dx=\int \left (e + f x\right )^{2} \sqrt {- d x + 1} \sqrt {d x + 1} \left (A + B x + C x^{2}\right )\, dx \] Input:
integrate((-d*x+1)**(1/2)*(d*x+1)**(1/2)*(f*x+e)**2*(C*x**2+B*x+A),x)
Output:
Integral((e + f*x)**2*sqrt(-d*x + 1)*sqrt(d*x + 1)*(A + B*x + C*x**2), x)
Time = 0.12 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.08 \[ \int \sqrt {1-d x} \sqrt {1+d x} (e+f x)^2 \left (A+B x+C x^2\right ) \, dx=-\frac {{\left (-d^{2} x^{2} + 1\right )}^{\frac {3}{2}} C f^{2} x^{3}}{6 \, d^{2}} + \frac {1}{2} \, \sqrt {-d^{2} x^{2} + 1} A e^{2} x + \frac {A e^{2} \arcsin \left (d x\right )}{2 \, d} - \frac {{\left (-d^{2} x^{2} + 1\right )}^{\frac {3}{2}} B e^{2}}{3 \, d^{2}} - \frac {2 \, {\left (-d^{2} x^{2} + 1\right )}^{\frac {3}{2}} A e f}{3 \, d^{2}} - \frac {{\left (-d^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (2 \, C e f + B f^{2}\right )} x^{2}}{5 \, d^{2}} - \frac {{\left (-d^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (C e^{2} + 2 \, B e f + A f^{2}\right )} x}{4 \, d^{2}} - \frac {{\left (-d^{2} x^{2} + 1\right )}^{\frac {3}{2}} C f^{2} x}{8 \, d^{4}} + \frac {\sqrt {-d^{2} x^{2} + 1} {\left (C e^{2} + 2 \, B e f + A f^{2}\right )} x}{8 \, d^{2}} + \frac {\sqrt {-d^{2} x^{2} + 1} C f^{2} x}{16 \, d^{4}} + \frac {{\left (C e^{2} + 2 \, B e f + A f^{2}\right )} \arcsin \left (d x\right )}{8 \, d^{3}} + \frac {C f^{2} \arcsin \left (d x\right )}{16 \, d^{5}} - \frac {2 \, {\left (-d^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (2 \, C e f + B f^{2}\right )}}{15 \, d^{4}} \] Input:
integrate((-d*x+1)^(1/2)*(d*x+1)^(1/2)*(f*x+e)^2*(C*x^2+B*x+A),x, algorith m="maxima")
Output:
-1/6*(-d^2*x^2 + 1)^(3/2)*C*f^2*x^3/d^2 + 1/2*sqrt(-d^2*x^2 + 1)*A*e^2*x + 1/2*A*e^2*arcsin(d*x)/d - 1/3*(-d^2*x^2 + 1)^(3/2)*B*e^2/d^2 - 2/3*(-d^2* x^2 + 1)^(3/2)*A*e*f/d^2 - 1/5*(-d^2*x^2 + 1)^(3/2)*(2*C*e*f + B*f^2)*x^2/ d^2 - 1/4*(-d^2*x^2 + 1)^(3/2)*(C*e^2 + 2*B*e*f + A*f^2)*x/d^2 - 1/8*(-d^2 *x^2 + 1)^(3/2)*C*f^2*x/d^4 + 1/8*sqrt(-d^2*x^2 + 1)*(C*e^2 + 2*B*e*f + A* f^2)*x/d^2 + 1/16*sqrt(-d^2*x^2 + 1)*C*f^2*x/d^4 + 1/8*(C*e^2 + 2*B*e*f + A*f^2)*arcsin(d*x)/d^3 + 1/16*C*f^2*arcsin(d*x)/d^5 - 2/15*(-d^2*x^2 + 1)^ (3/2)*(2*C*e*f + B*f^2)/d^4
Leaf count of result is larger than twice the leaf count of optimal. 1059 vs. \(2 (264) = 528\).
Time = 0.24 (sec) , antiderivative size = 1059, normalized size of antiderivative = 3.73 \[ \int \sqrt {1-d x} \sqrt {1+d x} (e+f x)^2 \left (A+B x+C x^2\right ) \, dx=\text {Too large to display} \] Input:
integrate((-d*x+1)^(1/2)*(d*x+1)^(1/2)*(f*x+e)^2*(C*x^2+B*x+A),x, algorith m="giac")
Output:
1/240*(120*(sqrt(d*x + 1)*(d*x - 2)*sqrt(-d*x + 1) - 2*arcsin(1/2*sqrt(2)* sqrt(d*x + 1)))*A*d^4*e^2 + 240*(sqrt(d*x + 1)*sqrt(-d*x + 1) + 2*arcsin(1 /2*sqrt(2)*sqrt(d*x + 1)))*A*d^4*e^2 + 40*(((2*d*x - 5)*(d*x + 1) + 9)*sqr t(d*x + 1)*sqrt(-d*x + 1) + 6*arcsin(1/2*sqrt(2)*sqrt(d*x + 1)))*B*d^3*e^2 + 120*(sqrt(d*x + 1)*(d*x - 2)*sqrt(-d*x + 1) - 2*arcsin(1/2*sqrt(2)*sqrt (d*x + 1)))*B*d^3*e^2 + 80*(((2*d*x - 5)*(d*x + 1) + 9)*sqrt(d*x + 1)*sqrt (-d*x + 1) + 6*arcsin(1/2*sqrt(2)*sqrt(d*x + 1)))*A*d^3*e*f + 240*(sqrt(d* x + 1)*(d*x - 2)*sqrt(-d*x + 1) - 2*arcsin(1/2*sqrt(2)*sqrt(d*x + 1)))*A*d ^3*e*f + 10*(((2*(3*d*x - 10)*(d*x + 1) + 43)*(d*x + 1) - 39)*sqrt(d*x + 1 )*sqrt(-d*x + 1) - 18*arcsin(1/2*sqrt(2)*sqrt(d*x + 1)))*C*d^2*e^2 + 40*(( (2*d*x - 5)*(d*x + 1) + 9)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 6*arcsin(1/2*sqr t(2)*sqrt(d*x + 1)))*C*d^2*e^2 + 20*(((2*(3*d*x - 10)*(d*x + 1) + 43)*(d*x + 1) - 39)*sqrt(d*x + 1)*sqrt(-d*x + 1) - 18*arcsin(1/2*sqrt(2)*sqrt(d*x + 1)))*B*d^2*e*f + 80*(((2*d*x - 5)*(d*x + 1) + 9)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 6*arcsin(1/2*sqrt(2)*sqrt(d*x + 1)))*B*d^2*e*f + 10*(((2*(3*d*x - 10)*(d*x + 1) + 43)*(d*x + 1) - 39)*sqrt(d*x + 1)*sqrt(-d*x + 1) - 18*arcs in(1/2*sqrt(2)*sqrt(d*x + 1)))*A*d^2*f^2 + 40*(((2*d*x - 5)*(d*x + 1) + 9) *sqrt(d*x + 1)*sqrt(-d*x + 1) + 6*arcsin(1/2*sqrt(2)*sqrt(d*x + 1)))*A*d^2 *f^2 + 4*(((2*(3*(4*d*x - 17)*(d*x + 1) + 133)*(d*x + 1) - 295)*(d*x + 1) + 195)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 90*arcsin(1/2*sqrt(2)*sqrt(d*x + ...
Time = 30.93 (sec) , antiderivative size = 2920, normalized size of antiderivative = 10.28 \[ \int \sqrt {1-d x} \sqrt {1+d x} (e+f x)^2 \left (A+B x+C x^2\right ) \, dx=\text {Too large to display} \] Input:
int((e + f*x)^2*(1 - d*x)^(1/2)*(d*x + 1)^(1/2)*(A + B*x + C*x^2),x)
Output:
- ((((1 - d*x)^(1/2) - 1)^8*((4928*B*f^2)/3 + (512*B*d^2*e^2)/3))/((d*x + 1)^(1/2) - 1)^8 - (((1 - d*x)^(1/2) - 1)^14*((1408*B*f^2)/3 - (32*B*d^2*e^ 2)/3))/((d*x + 1)^(1/2) - 1)^14 - (((1 - d*x)^(1/2) - 1)^6*((1408*B*f^2)/3 - (32*B*d^2*e^2)/3))/((d*x + 1)^(1/2) - 1)^6 + (((1 - d*x)^(1/2) - 1)^12* ((4928*B*f^2)/3 + (512*B*d^2*e^2)/3))/((d*x + 1)^(1/2) - 1)^12 - (((1 - d* x)^(1/2) - 1)^10*((11008*B*f^2)/5 - 304*B*d^2*e^2))/((d*x + 1)^(1/2) - 1)^ 10 + (64*B*f^2*((1 - d*x)^(1/2) - 1)^4)/((d*x + 1)^(1/2) - 1)^4 + (64*B*f^ 2*((1 - d*x)^(1/2) - 1)^16)/((d*x + 1)^(1/2) - 1)^16 + (8*B*d^2*e^2*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (8*B*d^2*e^2*((1 - d*x)^(1/2) - 1)^18)/((d*x + 1)^(1/2) - 1)^18 + (33*B*d*e*f*((1 - d*x)^(1/2) - 1)^3)/ ((d*x + 1)^(1/2) - 1)^3 - (204*B*d*e*f*((1 - d*x)^(1/2) - 1)^5)/((d*x + 1) ^(1/2) - 1)^5 + (204*B*d*e*f*((1 - d*x)^(1/2) - 1)^7)/((d*x + 1)^(1/2) - 1 )^7 + (442*B*d*e*f*((1 - d*x)^(1/2) - 1)^9)/((d*x + 1)^(1/2) - 1)^9 - (442 *B*d*e*f*((1 - d*x)^(1/2) - 1)^11)/((d*x + 1)^(1/2) - 1)^11 - (204*B*d*e*f *((1 - d*x)^(1/2) - 1)^13)/((d*x + 1)^(1/2) - 1)^13 + (204*B*d*e*f*((1 - d *x)^(1/2) - 1)^15)/((d*x + 1)^(1/2) - 1)^15 - (33*B*d*e*f*((1 - d*x)^(1/2) - 1)^17)/((d*x + 1)^(1/2) - 1)^17 + (B*d*e*f*((1 - d*x)^(1/2) - 1)^19)/(( d*x + 1)^(1/2) - 1)^19 - (B*d*e*f*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1))/(d^4 + (10*d^4*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (4 5*d^4*((1 - d*x)^(1/2) - 1)^4)/((d*x + 1)^(1/2) - 1)^4 + (120*d^4*((1 -...
Time = 0.16 (sec) , antiderivative size = 577, normalized size of antiderivative = 2.03 \[ \int \sqrt {1-d x} \sqrt {1+d x} (e+f x)^2 \left (A+B x+C x^2\right ) \, dx=\frac {-240 \mathit {asin} \left (\frac {\sqrt {-d x +1}}{\sqrt {2}}\right ) a \,d^{4} e^{2}-60 \mathit {asin} \left (\frac {\sqrt {-d x +1}}{\sqrt {2}}\right ) a \,d^{2} f^{2}-60 \mathit {asin} \left (\frac {\sqrt {-d x +1}}{\sqrt {2}}\right ) c \,d^{2} e^{2}-32 \sqrt {d x +1}\, \sqrt {-d x +1}\, c \,d^{3} e f \,x^{2}+120 \sqrt {d x +1}\, \sqrt {-d x +1}\, a \,d^{5} e^{2} x +60 \sqrt {d x +1}\, \sqrt {-d x +1}\, a \,d^{5} f^{2} x^{3}-160 \sqrt {d x +1}\, \sqrt {-d x +1}\, a \,d^{3} e f -30 \sqrt {d x +1}\, \sqrt {-d x +1}\, a \,d^{3} f^{2} x +80 \sqrt {d x +1}\, \sqrt {-d x +1}\, b \,d^{5} e^{2} x^{2}+48 \sqrt {d x +1}\, \sqrt {-d x +1}\, b \,d^{5} f^{2} x^{4}-16 \sqrt {d x +1}\, \sqrt {-d x +1}\, b \,d^{3} f^{2} x^{2}+60 \sqrt {d x +1}\, \sqrt {-d x +1}\, c \,d^{5} e^{2} x^{3}+40 \sqrt {d x +1}\, \sqrt {-d x +1}\, c \,d^{5} f^{2} x^{5}-30 \sqrt {d x +1}\, \sqrt {-d x +1}\, c \,d^{3} e^{2} x -10 \sqrt {d x +1}\, \sqrt {-d x +1}\, c \,d^{3} f^{2} x^{3}-64 \sqrt {d x +1}\, \sqrt {-d x +1}\, c d e f -15 \sqrt {d x +1}\, \sqrt {-d x +1}\, c d \,f^{2} x -120 \mathit {asin} \left (\frac {\sqrt {-d x +1}}{\sqrt {2}}\right ) b \,d^{2} e f +160 \sqrt {d x +1}\, \sqrt {-d x +1}\, a \,d^{5} e f \,x^{2}+120 \sqrt {d x +1}\, \sqrt {-d x +1}\, b \,d^{5} e f \,x^{3}-60 \sqrt {d x +1}\, \sqrt {-d x +1}\, b \,d^{3} e f x +96 \sqrt {d x +1}\, \sqrt {-d x +1}\, c \,d^{5} e f \,x^{4}-30 \mathit {asin} \left (\frac {\sqrt {-d x +1}}{\sqrt {2}}\right ) c \,f^{2}-80 \sqrt {d x +1}\, \sqrt {-d x +1}\, b \,d^{3} e^{2}-32 \sqrt {d x +1}\, \sqrt {-d x +1}\, b d \,f^{2}}{240 d^{5}} \] Input:
int((-d*x+1)^(1/2)*(d*x+1)^(1/2)*(f*x+e)^2*(C*x^2+B*x+A),x)
Output:
( - 240*asin(sqrt( - d*x + 1)/sqrt(2))*a*d**4*e**2 - 60*asin(sqrt( - d*x + 1)/sqrt(2))*a*d**2*f**2 - 120*asin(sqrt( - d*x + 1)/sqrt(2))*b*d**2*e*f - 60*asin(sqrt( - d*x + 1)/sqrt(2))*c*d**2*e**2 - 30*asin(sqrt( - d*x + 1)/ sqrt(2))*c*f**2 + 120*sqrt(d*x + 1)*sqrt( - d*x + 1)*a*d**5*e**2*x + 160*s qrt(d*x + 1)*sqrt( - d*x + 1)*a*d**5*e*f*x**2 + 60*sqrt(d*x + 1)*sqrt( - d *x + 1)*a*d**5*f**2*x**3 - 160*sqrt(d*x + 1)*sqrt( - d*x + 1)*a*d**3*e*f - 30*sqrt(d*x + 1)*sqrt( - d*x + 1)*a*d**3*f**2*x + 80*sqrt(d*x + 1)*sqrt( - d*x + 1)*b*d**5*e**2*x**2 + 120*sqrt(d*x + 1)*sqrt( - d*x + 1)*b*d**5*e* f*x**3 + 48*sqrt(d*x + 1)*sqrt( - d*x + 1)*b*d**5*f**2*x**4 - 80*sqrt(d*x + 1)*sqrt( - d*x + 1)*b*d**3*e**2 - 60*sqrt(d*x + 1)*sqrt( - d*x + 1)*b*d* *3*e*f*x - 16*sqrt(d*x + 1)*sqrt( - d*x + 1)*b*d**3*f**2*x**2 - 32*sqrt(d* x + 1)*sqrt( - d*x + 1)*b*d*f**2 + 60*sqrt(d*x + 1)*sqrt( - d*x + 1)*c*d** 5*e**2*x**3 + 96*sqrt(d*x + 1)*sqrt( - d*x + 1)*c*d**5*e*f*x**4 + 40*sqrt( d*x + 1)*sqrt( - d*x + 1)*c*d**5*f**2*x**5 - 30*sqrt(d*x + 1)*sqrt( - d*x + 1)*c*d**3*e**2*x - 32*sqrt(d*x + 1)*sqrt( - d*x + 1)*c*d**3*e*f*x**2 - 1 0*sqrt(d*x + 1)*sqrt( - d*x + 1)*c*d**3*f**2*x**3 - 64*sqrt(d*x + 1)*sqrt( - d*x + 1)*c*d*e*f - 15*sqrt(d*x + 1)*sqrt( - d*x + 1)*c*d*f**2*x)/(240*d **5)